Finite Group Databasebeta

← #20283936

Subgroups — #20283936

Order 6784.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D53C211yesmaximal, characteristic, complemented
2C2×#13133240C211yesmaximal, characteristic, complemented
3C2×#13133238C211yesmaximal, characteristic, complemented
4#16811136C211yesmaximal, characteristic, complemented
5#16810783C211yesmaximal, characteristic, complemented
6#16810777C211yesmaximal, characteristic, complemented
7#16810772C211yesmaximal, characteristic, complemented
8#16810770C211yesmaximal, characteristic, complemented
9#16810768C211yesmaximal, characteristic, complemented
10#16810763C211yesmaximal, characteristic, complemented
11#16810758C211yesmaximal, characteristic, complemented
12#16810618C211yesmaximal, characteristic, complemented
13#16810616C211yesmaximal, characteristic, complemented
14C22⋊C8×D53C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719949C2211yescharacteristic, complemented
18C2×#11719945—21no—
19C2×#11719945C2211yescharacteristic
20C2×#11719947—21no—
21C2×#11719947C2211yescharacteristic
22#41×D53—24no—
23#41×D53—24no—
24C2×Q8×D53C2211yescharacteristic
25C2×D4×D53—21no—
26C2×D4×D53C2211yescharacteristic
27C22×C4×D53—21no—
28C22×C4×D53C2211yescharacteristic
29#13133240—24no—
30#13133238—24no—
31C2×#11719925C2211yescharacteristic, complemented
32C2×#11719913C2211yescharacteristic, complemented
33C2×#11719924C2211yescharacteristic, complemented
34C2×#11719924C2211yescharacteristic, complemented
35C2×D424C2211yescharacteristic, complemented
36C2×#11719922C2211yescharacteristic, complemented
37C2×C8×D53C2211yescharacteristic, complemented
38C2×C8×D53C2211yescharacteristic, complemented
39#13133202—21no—
40#13133193C2211yescharacteristic
41#13133192C2211yescharacteristic
42#13133191C2211yescharacteristic, complemented
43#13133187—21no—
44#13133179—21no—
45#13133177—21no—
46#13133175C2211yescharacteristic, complemented
47#13133142C2211yescharacteristic, complemented
48#13133141C2211yescharacteristic, complemented
49#13133140C2211yescharacteristic, complemented
50#13133139C2211yescharacteristic, complemented
51#13133138C2211yescharacteristic, complemented
52#13133137C2211yescharacteristic, complemented
53C22⋊#88322C2211yescharacteristic, complemented
54#13133129C2211yescharacteristic
55#13133129C2211yescharacteristic
56C2×#41×C53C2211yescharacteristic
57C2×Q16×C53C2211yescharacteristic, complemented
58C2×SD16×C53C2211yescharacteristic, complemented
59C22⋊Q8×C53C2211yescharacteristic, complemented
60#103×C53C2211yescharacteristic, complemented
61#102×C53C2211yescharacteristic, complemented
62C22⋊C8×C53C2211yescharacteristic, complemented
63C23×D53—21no—
64C23×D53D411yescharacteristic
65#41×C53—22no—
66#41×C53—22no—
67C2×Q8×C53C2311yescharacteristic
68C2×Q8×C53C2311yescharacteristic
69C2×D4×C53—21no—
70C2×D4×C53C2311yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53C2311yescharacteristic
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C22×Dic53—21no—
78C22×Dic53D411yescharacteristic
79#11719949—22no—
80#11719949—22no—
81#11719949—22no—
82#11719949D414yescomplemented
83Q8×D53—22no—
84Q8×D53—22no—
85#11719947—28no—
86#11719947—22no—
87#11719947D414yescomplemented
88D4×D53—28no—
89D4×D53—22no—
90D4×D53—22no—
91#11719945—28no—
92#11719945—22no—
93#11719945—22no—
94C2×D212—21no—
95C2×D212—21no—
96C2×D212—21no—
97C2×D212C2311yescharacteristic
98C2×D212C2311yescharacteristic
99C2×C4×D53—28no—
100C2×C4×D53—21no—
101C2×C4×D53—21no—
102C2×C4×D53—21no—
103C2×C4×D53—22no—
104C2×C4×D53—22no—
105C2×C4×D53—21no—
106C2×C4×D53D411yescharacteristic
107C2×C4×D53D411yescharacteristic
108C2×C4×D53C2311yescharacteristic
109C2×C4×D53C2311yescharacteristic
110C2×Dic106—21no—
111C2×Dic106C2311yescharacteristic
112Q16×C53—22no—
113SD16×C53—22no—
114C2×C8×C53C2311yescharacteristic
115C2×C8×C53C2311yescharacteristic
116C4⋊C4×C53—21no—
117C4⋊C4×C53C2311yescharacteristic
118C22⋊C4×C53—21no—
119#11719925—22no—
120#11719924—22no—
121#11719924—22no—
122#11719922—22no—
123#11719921—21no—
124#11719921—21no—
125#11719921—21no—
126#11719921—21no—
127C4⋊Dic53C2311yescharacteristic
128#11719919—21no—
129C4×Dic53—21no—
130C2×#88322C2311yescharacteristic
131C2×#88322C2311yescharacteristic
132D424—22no—
133#11719913—22no—
134C8×D53—22no—
135C8×D53—22no—
136C23×C53—21no—
137C23×C53C2×D411yescharacteristic
138C22×D53—24no—
139C22×D53—21no—
140C22×D53—22no—
141C22×D53—22no—
142C22×D53—21no—
143C22×D53—41no—
144C22×D53—42no—
145C22×D53—42no—
146C22×D53C2×D411yescharacteristic
147C22×D53C2×D411yescharacteristic
148Q8×C53—21no—
149Q8×C53—21no—
150Q8×C53—21no—
151Q8×C53C2×D412yes—
152D4×C53—24no—
153D4×C53—21no—
154D4×C53C2×D412yes—
155C2×C4×C53—24no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53C2×D411yescharacteristic
162C2×C4×C53C2411yescharacteristic
163C22⋊D53—24no—
164C22⋊D53—24no—
165C22⋊D53—24no—
166C22⋊D53—42no—
167C2×Dic53—21no—
168C2×Dic53—22no—
169C2×Dic53—24no—
170C2×Dic53—21no—
171C2×Dic53C2×D411yescharacteristic
172C2×Dic53C2×D411yescharacteristic
173D212—24no—
174D212—21no—
175D212—21no—
176D212—21no—
177D212—42no—
178D212—42no—
179D212C2×D412yes—
180C4×D53—24no—
181C4×D53—24no—
182C4×D53—22no—
183C4×D53—21no—
184C4×D53—21no—
185C4×D53—22no—
186C4×D53—24no—
187C4×D53—42no—
188C4×D53C2×D412yes—
189C4×D53C2×D412yes—
190Dic106—24no—
191Dic106—21no—
192Dic106C2×D412yes—
193C8×C53—21no—
194C8×C53—21no—
195#88322—21no—
196#88322—21no—
197C22×C53—21no—
198C22×C53—22no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53C22×D411yescharacteristic
204C2×D53—21no—
205C2×D53—21no—
206C2×D53—21no—
207C2×D53—21no—
208C2×D53—22no—
209C2×D53—44no—
210C2×D53—48no—
211C2×D53—41no—
212C2×D53—41no—
213C2×D53—41no—
214C2×D53—41no—
215C2×D53—41no—
216C2×D53#12012yes—
217C2×D53#13512yescomplemented
218C4×C53—22no—
219C4×C53—21no—
220C4×C53—41no—
221C4×C53—41no—
222C4×C53C22×D411yescharacteristic
223C4×C53C22×D411yescharacteristic
224Dic53—21no—
225Dic53—22no—
226Dic53—41no—
227Dic53#12012yes—
228#3002—531nomaximal, Sylow
229C2×C53—41no—
230C2×C53—42no—
231C2×C53C2×#12011yescharacteristic
232C2×C53C2×#13511yescharacteristic
233C2×C53#57511yescharacteristic
234D53—22no—
235D53—22no—
236D53—42no—
237D53—81no—
238C22×#41—531no—
239C2×#135—531no—
240C2×#135—531no—
241#525—531no—
242#452—531no—
243#452D5311yescharacteristic, complemented
244#451—531no—
245#450—531no—
246#449—531no—
247#449—531no—
248#448—531no—
249#447—531no—
250#416—531no—
251#416—531no—
252C2×C22⋊C8—531no—
253C53#300211yescharacteristic, complemented, Sylow
254C2×#41—531no—
255C2×#41—531no—
256C2×#41—531no—
257C2×#41—531no—
258C2×#41—1061no—
259C2×#41—1061no—
260C2×#41—1064no—
261C2×#41—1064no—
262C2×#41C2×D5311yescharacteristic
263C22×Q8—531no—
264C22×D4—531no—
265C22×D4—1061no—
266C23×C4—531no—
267C23×C4—1061no—
268#135—1064no—
269#135—1064no—
270C2×Q16—531no—
271C2×Q16C2×D5311yescharacteristic, complemented
272C2×SD16—531no—
273C2×SD16—531no—
274C2×SD16—531no—
275C2×SD16C2×D5311yescharacteristic, complemented
276C2×D8—531no—
277C2×D8—531no—
278C22×C8—531no—
279C22×C8—531no—
280#123—1061no—
281C22⋊Q8—531no—
282C22⋊Q8C2×D5311yescharacteristic, complemented
283#121—531no—
284#121—531no—
285#120—1061no—
286C4×D4—1061no—
287C4×D4—1061no—
288#117—531no—
289#103—531no—
290#103—531no—
291#103—531no—
292#103C2×D5311yescharacteristic, complemented
293#102—531no—
294#102—531no—
295#102—531no—
296#102C2×D5311yescharacteristic, complemented
297C22⋊C8—531no—
298C22⋊C8—531no—
299C22⋊C8—531no—
300C22⋊C8C2×D5311yescharacteristic, complemented
301C24—531no—
302C24—1061no—
303#41—22no—
304#41—22no—
305#41—534no—
306#41—534no—
307#41—1062no—
308#41—1062no—
309#41—1068no—
310#41—1062no—
311#41—1062no—
312#41—1062no—
313#41—1062no—
314#41—1068no—
315C2×Q8—531no—
316C2×Q8—1061no—
317C2×Q8—1062no—
318C2×Q8—1062no—
319C2×Q8C22×D5311yescharacteristic
320C2×Q8C22×D5311yescharacteristic
321C2×D4—21no—
322C2×D4—531no—
323C2×D4—531no—
324C2×D4—1061no—
325C2×D4—1061no—
326C2×D4—1061no—
327C2×D4—1061no—
328C2×D4—1062no—
329C2×D4—1068no—
330C2×D4—1061no—
331C2×D4—1061no—
332C2×D4—1062no—
333C2×D4—1061no—
334C2×D4C22×D5311yescharacteristic
335C22×C4—21no—
336C22×C4—531no—
337C22×C4—531no—
338C22×C4—531no—
339C22×C4—531no—
340C22×C4—531no—
341C22×C4—1062no—
342C22×C4—1062no—
343C22×C4—1061no—
344C22×C4—1061no—
345C22×C4—1061no—
346C22×C4—1068no—
347C22×C4—1061no—
348C22×C4—1061no—
349C22×C4C22×D5311yescharacteristic
350Q16—22no—
351Q16—1062no—
352SD16—22no—
353SD16—1062no—
354SD16—1062no—
355SD16—1062no—
356D8—1062no—
357D8—1062no—
358C2×C8—531no—
359C2×C8—531no—
360C2×C8—1062no—
361C2×C8—1062no—
362C2×C8C22×D5311yescharacteristic
363C2×C8C22×D5311yescharacteristic
364C4⋊C4—21no—
365C4⋊C4—531no—
366C4⋊C4—1061no—
367C4⋊C4C22×D5311yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—1061no—
370C22⋊C4—1061no—
371C22⋊C4—1061no—
372C22⋊C4—1061no—
373C42—1061no—
374C23—21no—
375C23—531no—
376C23—531no—
377C23—1062no—
378C23—1062no—
379C23—1064no—
380C23—1061no—
381C23—1061no—
382C23—2121no—
383C23—2122no—
384C23—2122no—
385C23D4×D5311yescharacteristic
386Q8—21no—
387Q8—21no—
388Q8—21no—
389Q8—532no—
390Q8—1064no—
391Q8—1061no—
392Q8D4×D5312yes—
393D4—24no—
394D4—21no—
395D4—532no—
396D4—1064no—
397D4—1064no—
398D4—1064no—
399D4—1064no—
400D4—1061no—
401D4—1061no—
402D4—1061no—
403D4—2122no—
404D4—2122no—
405D4—2122no—
406D4D4×D5312yes—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—531no—
414C2×C4—532no—
415C2×C4—532no—
416C2×C4—531no—
417C2×C4—1062no—
418C2×C4—1064no—
419C2×C4—1062no—
420C2×C4—1064no—
421C2×C4—1064no—
422C2×C4—1062no—
423C2×C4—1064no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1061no—
427C2×C4—1061no—
428C2×C4—2122no—
429C2×C4D4×D5311yescharacteristic
430C2×C4C23×D5311yescharacteristic
431C8—21no—
432C8—21no—
433C8—1061no—
434C8—1061no—
435C22—21no—
436C22—22no—
437C22—21no—
438C22—21no—
439C22—41no—
440C22—41no—
441C22—532no—
442C22—532no—
443C22—1061no—
444C22—1061no—
445C22—1061no—
446C22—1061no—
447C22—1062no—
448C22—2124no—
449C22—2128no—
450C22—2121no—
451C22—2121no—
452C22—2121no—
453C22—2121no—
454C22—2121no—
455C22C2×D4×D5311yescharacteristic
456C4—21no—
457C4—22no—
458C4—41no—
459C4—41no—
460C4—532no—
461C4—1061no—
462C4—1062no—
463C4—2121no—
464C4C2×D4×D5311yescharacteristic
465C4C2×D4×D5311yescharacteristic
466C2—41no—
467C2—42no—
468C2—1062no—
469C2—1062no—
470C2—4241no—
471C2—2122no—
472C2#120×D5311yescharacteristic
473C2#135×D5311yescharacteristic
474C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3002
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#41×D53C2semidirect
2. C2×#13133240C2semidirect
3. C2×#13133238C2semidirect
4. #16811136C2semidirect
5. #16810783C2semidirect
6. #16810777C2semidirect
7. #16810772C2semidirect
8. #16810770C2semidirect
9. #16810768C2semidirect
10. #16810763C2semidirect
11. #16810758C2semidirect
12. #16810618C2semidirect
13. #16810616C2semidirect
14. C22⋊C8×D53C2semidirect
15. C53×#452C2semidirect
17. C2×#11719949C22semidirect
31. C2×#11719925C22semidirect
32. C2×#11719913C22semidirect
33. C2×#11719924C22semidirect
34. C2×#11719924C22semidirect
35. C2×D424C22semidirect
36. C2×#11719922C22semidirect
37. C2×C8×D53C22semidirect
38. C2×C8×D53C22semidirect
42. #13133191C22semidirect
46. #13133175C22semidirect
47. #13133142C22semidirect
48. #13133141C22semidirect
49. #13133140C22semidirect
50. #13133139C22semidirect
51. #13133138C22semidirect
52. #13133137C22semidirect
53. C22⋊#88322C22semidirect
57. C2×Q16×C53C22semidirect
58. C2×SD16×C53C22semidirect
59. C22⋊Q8×C53C22semidirect
60. #103×C53C22semidirect
61. #102×C53C22semidirect
62. C22⋊C8×C53C22semidirect
82. #11719949D4semidirect
87. #11719947D4semidirect
217. C2×D53#135semidirect
243. #452D53semidirect
253. C53#3002semidirect
271. C2×Q16C2×D53semidirect
275. C2×SD16C2×D53semidirect
282. C22⋊Q8C2×D53semidirect
292. #103C2×D53semidirect
296. #102C2×D53semidirect
300. C22⋊C8C2×D53semidirect